We apply the methodology of Markov chains to analyzing a baseball offense. We first review Cover and Keilers's Offensive Earned -Run Average, a method of evaluating individual players. We then extend their work by presenting a method of modeling arbitrary lineups. Using this information together with a calculation of how often each batter leads off an inning, we can calculate the expected runs scored per game for any lineup. In contrast to previous work on lineup selection, which has used empirical data or simulation, we derive exact solutions to explicit mathematical models. The method can also be used to evaluate an individual player, not just in isolation but
in the context of a specific lineup. We apply these methods
to data from the 1993 season and present computer
code for their computation.